Research on characteristics of noise-perturbed M–J sets based on equipotential point algorithm

AbstractAs the classical ones among the fractal sets, Julia set (abbreviated as J set) and Mandelbrot set (abbreviated as M set) have been explored widely in recent years. In this study, J set and M set under additive noise perturbation and multiplicative noise perturbation are created by equipotential point algorithm. Changes of the J set and M set under random noise perturbation as well as the close correlation between them are studied. Experimental results show that either additive noise perturbation or multiplicative noise perturbation may cause dramatic changes on J set. On the other hand, when the M set is perturbed by additive noise, it almost changes nothing but its position; when the M set is perturbed by multiplicative noise, its inner structures change with the stabilized areas shrinking, but it keeps the symmetry with respect to X axis. In addition, the J set and the M set still share the same stabilized periodic point in spite of noise perturbation

On numerical approximations of the area of the generalized Mandelbrot sets

In the present work, the area of the generalized Mandelbrot sets is defined as the double limit of the areas of the plotted generalized Mandelbrot sets in a given square lattice, using the finite escape algorithm, while the lattice resolution and the number of iteration counts, used to plot them, tends to infinity. The asymptotic behavior of the areas of the generalized Mandelbrot sets in terms of their degree growth is investigated. Finally, numerical approximations of the area of the Mandelbrot set are proposed by using tools from regression analysis. (C) 2013 Elsevier Inc. All rights reserved